In comments following this answer of mine to a related question, Users ssdecontrol and Glen_b asked
whether joint normality of $X$ and $Y$ is necessary for asserting the
normality of the sum $X+Y$? That joint normality is sufficient is,
of course, well-known. This supplemental question was not addressed
there, and is perhaps worth considering in its own right.
Since joint normality implies marginal normality, I ask
Do there exist normal random variables $X$ and $Y$ such that
$X+Y$ is a normal random variable, but $X$ and $Y$ are not
jointly normal random variables?
If $X$ and $Y$ are not required to have normal distributions,
then it is easy to find such normal random variables. One example
can be found in my previous answer (link is given above).
I believe that the answer to the highlighted question above
is Yes, and have posted (what I think is) an example as an answer
to this question.
Let $U,V$ be iid $N(0,1)$.
Now transform $(U,V) \to (X,Y)$ as follows:
In the first quadrant (i.e. $U>0,V>0$) let $X=\max(U,V)$ and $Y = \min(U,V)$.
For the other quadrants, rotate this mapping about the origin.
The resulting bivariate distribution looks like (seen from above):
— the purple represents regions with doubled probability and the white regions are ones with no probability. The black circles are contours of constant density (everywhere on the circle for $(U,V)$, but within each colored region for $(X,Y)$).
By symmetry both $X$ and $Y$ are standard normal (looking down a vertical line or along a horizontal line there’s a purple point for every white one which we can regard as being flipped across the axis the horizontal or vertical line crosses)
but $(X,Y)$ are clearly not bivariate normal, and
$X+Y = U+V$ which is $\sim N(0,2)$ (equivalently, look along lines of constant $X+Y$ and see that we have symmetry similar to that we discussed in 1., but this time about the $Y=X$ line)